Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{p^2 - 4p - 21}{-4p + 20} \div \dfrac{p + 3}{p - 5} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{p^2 - 4p - 21}{-4p + 20} \times \dfrac{p - 5}{p + 3} $ First factor the quadratic. $z = \dfrac{(p + 3)(p - 7)}{-4p + 20} \times \dfrac{p - 5}{p + 3} $ Then factor out any other terms. $z = \dfrac{(p + 3)(p - 7)}{-4(p - 5)} \times \dfrac{p - 5}{p + 3} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac{ (p + 3)(p - 7) \times (p - 5) } { -4(p - 5) \times (p + 3) } $ $z = \dfrac{ (p + 3)(p - 7)(p - 5)}{ -4(p - 5)(p + 3)} $ Notice that $(p - 5)$ and $(p + 3)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac{ \cancel{(p + 3)}(p - 7)(p - 5)}{ -4(p - 5)\cancel{(p + 3)}} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $z = \dfrac{ \cancel{(p + 3)}(p - 7)\cancel{(p - 5)}}{ -4\cancel{(p - 5)}\cancel{(p + 3)}} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $z = \dfrac{p - 7}{-4} $ $z = \dfrac{-(p - 7)}{4} ; \space p \neq -3 ; \space p \neq 5 $